### Nuprl Lemma : div_preserves_le

`∀[a,b:ℤ]. ∀[n:ℕ+].  ((a ≤ b) `` ((a ÷ n) ≤ (b ÷ n)))`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` le: `A ≤ B` implies: `P `` Q` divide: `n ÷ m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` nequal: `a ≠ b ∈ T ` nat_plus: `ℕ+` not: `¬A` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` uiff: `uiff(P;Q)` nat: `ℕ` less_than: `a < b` squash: `↓T` int_lower: `{...i}` gt: `i > j` ge: `i ≥ j `
Lemmas referenced :  int_term_value_minus_lemma itermMinus_wf div_bounds_2 div_bounds_1 rem_bounds_2 rem_bounds_1 false_wf int_term_value_subtract_lemma itermSubtract_wf subtract-is-int-iff add-is-int-iff div_rem_sum2 add-commutes one-mul mul-commutes mul-distributes nequal_wf less_than_wf subtype_rel_sets equal_wf nat_plus_subtype_nat mul_preserves_le nat_plus_wf less_than'_wf le_wf int_term_value_add_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma itermAdd_wf intformle_wf intformnot_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_and_lemma intformless_wf itermConstant_wf itermVar_wf intformeq_wf intformand_wf satisfiable-full-omega-tt nat_plus_properties decidable__le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin divideEquality because_Cache isectElimination hypothesisEquality hypothesis setElimination rename natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll unionElimination addEquality productElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry applyEquality multiplyEquality setEquality independent_functionElimination remainderEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed dependent_set_memberEquality imageElimination minusEquality

Latex:
\mforall{}[a,b:\mBbbZ{}].  \mforall{}[n:\mBbbN{}\msupplus{}].    ((a  \mleq{}  b)  {}\mRightarrow{}  ((a  \mdiv{}  n)  \mleq{}  (b  \mdiv{}  n)))

Date html generated: 2016_05_14-AM-07_24_32
Last ObjectModification: 2016_01_14-PM-10_03_07

Theory : int_2

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